Density of Elliptic Curves over Number Fields with Prescribed Torsion Subgroups

Let $K$ be a number field. For positive integers $m$ and $n$ such that $m\mid n$, we let ${\mathcal{S}}_{m,n}$ be the set of elliptic curves $E\mathrm{/}K$ defined over $K$ such that $E(K{)}_{\mathrm{tors}}\supseteq \mathcal{T}\cong \mathbb{Z}\mathrm{/}m\mathbb{Z}\times \mathbb{Z}\mathrm{/}n\mathbb{Z}$. We prove that if the genus of the modular curve $X_1(m,n)$ is $0$, then `almost all’ $E\in {\mathcal{S}}_{m,n}$ satisfy that $E(K{)}_{\mathrm{tors}}=\mathcal{T}$, i.e., not larger than $\mathcal{T}$. In particular, if $m=n=1$, this result generalizes Duke’s theorem over $\mathbb{Q}$ to arbitrary number fields $K$ for the trivial torsion subgroup.